Light elements abundance in a static toy universe

[TrickyDicky]

As an exercise in cosmology-fiction (I actually got curious about this from an actual cosmology textbook problem), taking into account the stellar nuclear reactions that involve the fusion of hydrogen into helium, what would (roughly) be the proportion (in mass) between Hydrogen and He-4 in a static universe?
Would it resemble the proportions of primordial nucleosynthesis given that these are produced close to thermodynamic equilibrium?

 

[Chronos]

Chemical abundances in 'pristine' [primordial] gas clouds would be very difficult to explain if the universe was significantly more ancient than we suspect. It is not uncommon that what you don't see tells you as much or more than what you do see.

 

[TrickyDicky]

Chemical abundances in 'pristine' [primordial] gas clouds would be very difficult to explain if the universe was significantly more ancient than we suspect.

A static universe can't be more or less "ancient", it is symply time-invariant as a whole.
It is totally discarded by science so I'm not sure what your comment means. My question is purely theoretical.

 

[BillSaltLake]

Are you assuming that one begins with nothing but pure hydrogen, and that any transmutation was stellar in origin ?

 

[TrickyDicky]

Are you assuming that one begins with nothing but pure hydrogen, and that any transmutation was stellar in origin ?

Yes, just the stellar nuclear reactions, only in a static universe makes little sense to say what one begins with, since time is invariant.

 

[BillSaltLake]

Then ⁴He, ³He, and ²H would be much rarer than observed. Most of these isotopes would be inside active stars and would be consumed during the stars' lives. The initial collapse into stars would be different (I'm not sure of the details) because the initial gas would not be composed of a quarter ⁴He. I assume there would still be supernovae and metal enrichment, though. There would be probably 1/100 or less ⁴He than ¹H, and most of that locked up in stars. Lower-mass stars (< ~1.5 solar masses) will usually become composed for a limited period of mostly ⁴He in later life, but this He will eventually fuse to heavier elements.

 

[TrickyDicky]

Then ⁴He, ³He, and ²H would be much rarer than observed. Most of these isotopes would be inside active stars and would be consumed during the stars' lives. The initial collapse into stars would be different (I'm not sure of the details) because the initial gas would not be composed of a quarter ⁴He. I assume there would still be supernovae and metal enrichment, though. There would be probably 1/100 or less ⁴He than ¹H, and most of that locked up in stars. Lower-mass stars (< ~1.5 solar masses) will usually become composed for a limited period of mostly ⁴He in later life, but this He will eventually fuse to heavier elements.

I guess you are ignoring that there is no global evolution in the kind of scenario I'm talking about. There can only be a permanent equilibrium distribution. I'm asking what would that distribution be according to the stellar nuclear reactions and core temperature conditions. Maybe it's easier to think in terms of proportion of neutrons and protons in thermodynamical equilibrium at the stars core temperatures.

 

[BillSaltLake]

The meaning of steady state might be a problem. After a very long time, everything might be photons and neutrinos (after black hole evaporation, and after any possible proton decay with the resulting positrons annihilating the existing electrons). I still doubt that the He could ever be over 1% of the H.
I'm assuming that H was distributed (with some density fluctuations) uniformly at the same density as baryonic matter now, and then it interacted, while the expansion factor remained constant. However, in a true steady state model, nothing would change, and the He ratio wouldn't change. There is no a priori equlibrium ratio unless there is a high-temp period or perhaps proton decay.

 

[TrickyDicky]

The meaning of steady state might be a problem. After a very long time, everything might be photons and neutrinos (after black hole evaporation, and after any possible proton decay with the resulting positrons annihilating the existing electrons). I still doubt that the He could ever be over 1% of the H.
I'm assuming that H was distributed (with some density fluctuations) uniformly at the same density as baryonic matter now, and then it interacted, while the expansion factor remained constant. However, in a true steady state model, nothing would change, and the He ratio wouldn't change. There is no a priori equlibrium ratio unless there is a high-temp period or perhaps proton decay.

Steady state and static spacetime are different notions, the most known steady state universe , that of Gold, Bondi and Hoyle was an expanding spacetime, it wasn't static.
In static spacetimes there is no global change wrt time (although there may be locally).
There can be no periods nor "after a very long time"s. No expansion factor either and if you bother to look up some GR text no possibility for the existence of black holes, so no bh evaporation either. As anyone can see it is a completely unrealistic cosmology.

I'm only trying to learn something about stellar nuclear reactions in equilibrium and in a time invariant situation to contrast it with the factual primordial nucleosynthesis context and have a better understanding of it.

 

[TrickyDicky]

Being more specific how much would the Nn/Np ratio in the time invariant situation in star's cores differ from the neutron freeze-out ratio in primordial nucleosynthesis (of around 1/6-1/7)?

 

[twofish-quant]

In any sort of time invariant universe, everything would end up as iron. The difficulty is that with light elements, there isn't a chemical equilibrium. Everything wants to be iron, and there are no cosmologically significant reverse reactions to break down iron into lighter elements.

However, this has been known since the 1940's, and the idea behind the steady state model was that there was a "magic source" of hydrogen to replace anything that got burned to He4. However, if we lived in a universe in which the magic source of hydrogen kept H/He ratios constant, we'd see no deuterium or He3, and a lot more Carbon-12 and heavier elements.

Being more specific how much would the Nn/Np ratio in the time invariant situation in star's cores differ from the neutron freeze-out ratio in primordial nucleosynthesis (of around 1/6-1/7)?

Very different. He4 to C12 has 1:1 and heavier elements are have more neutrons than proton culminating in neutron stars which are all neutrons.

 

[TrickyDicky]

In any sort of time invariant universe, everything would end up as iron. The difficulty is that with light elements, there isn't a chemical equilibrium. Everything wants to be iron, and there are no cosmologically significant reverse reactions to break down iron into lighter elements.

However, this has been known since the 1940's, and the idea behind the steady state model was that there was a "magic source" of hydrogen to replace anything that got burned to He4. However, if we lived in a universe in which the magic source of hydrogen kept H/He ratios constant, we'd see no deuterium or He3, and a lot more Carbon-12 and heavier elements.

For some reason you guys keep mixing static with eternal steady state models, even though I tried to clarify the difference in a previous post.
In steady state models there is time dependency, it is a expanding model. In static spacetimes nothing "ends up", there must be just an equilibrium distribution related to temperature, density and mass difference of protons and neutrons but independent of time.

 

[phyzguy]

For some reason you guys keep mixing static with eternal steady state models, even though I tried to clarify the difference in a previous post.
In steady state models there is time dependency, it is a expanding model. In static spacetimes nothing "ends up", there must be just an equilibrium distribution related to temperature, density and mass difference of protons and neutrons but independent of time.

There can be no equlibrium as you envision it unless there is a two-way pathway. So in your hypothetical static universe you need to answer twofish's question about what moves things "uphill" from iron back to hydrogen. Otherwise the equilibrium distribution in the universe you are asking about is a universe filled with black holes and iron. The steady-state universe answers this by postulating continuous creation of hydrogen and expansion.

 

[TrickyDicky]

There can be no equlibrium as you envision it unless there is a two-way pathway. So in your hypothetical static universe you need to answer twofish's question about what moves things "uphill" from iron back to hydrogen. Otherwise the equilibrium distribution in the universe you are asking about is a universe filled with black holes and iron. The steady-state universe answers this by postulating continuous creation of hydrogen and expansion.

What moves things "uphill" is the fact that in a static universe both ways of the two-way path have the same probability by definition, or at least that is what time-reversible reactions seem to imply.

 

[mathal]

What moves things "uphill" is the fact that in a static universe both ways of the two-way path have the same probability by definition, or at least that is what time-reversible reactions seem to imply.

Clearly in the model you present there is no entropy and yes theoretically all reactions are time invariant so without entropy there would be no way to discern a 'direction' to time. To answer your question, in such a universe you can have whatever ratio of elements you like since all reactions are matched with their counterpart (or your universe won't remain static).
mathal

 

[TrickyDicky]

Clearly in the model you present there is no entropy and yes theoretically all reactions are time invariant so without entropy there would be no way to discern a 'direction' to time. To answer your question, in such a universe you can have whatever ratio of elements you like since all reactions are matched with their counterpart (or your universe won't remain static).
mathal

Thanks for your input mathal, finally someone sees what I mean.
You are right, that was in fact my initial thought, that in such a universe any ratio would be possible, but I'm trying to introduce some constraints in the form of typical stellar core temperature , pressure and density and supposing the usual stellar nucleosynthesis reactions would also work so that some equilibrium distribution can be given that would make more sense than some other.

 

[Chronos]

An interstellar medium enriched by stellar ejecta would be anything but static. A universe without entropy would be not even wrong, but, I naively suspect it would remain in its original state. A universe without BB nuclosynthesis still needs a source of hydrogen for primordial stars to form. Once the stellar formation process began, the ISM would be continuously enriched by stellar ejecta. I fail to see how that could ever achieve equilibrium. It would merely continue to be enriched until so heavily metallized it could no longer support stellar fusion. The answer seems to be stars would not form in a universe without entropy.

 

[mathal]

An interstellar medium enriched by stellar ejecta would be anything but static. A universe without entropy would be not even wrong, but, I naively suspect it would remain in its original state.



A universe without BB nuclosynthesis still needs a source of hydrogen for primordial stars to form. Once the stellar formation process began, the ISM would be continuously enriched by stellar ejecta. I fail to see how that could ever achieve equilibrium. It would merely continue to be enriched until so heavily metallized it could no longer support stellar fusion. The answer seems to be stars would not form in a universe without entropy.

Exactly. This is merely a thought experiment. Objects like stars from our universe are not static- they age, things change, proportions of elements change with time. They couldn't exist in the form they have here. I find it hard to conceive of such a universe, the interrelationship of the laws that govern this universe are not workable without entropy. It is not that it is a physical law, merely that it is a consequence of the laws we operate under (in particular gravity).

You are thinking of primordial stars- our universe. The static universe TrickyDicky is presenting requires they just be here timelessly. An impossible requirement from my understanding of physics.
mathal

 

[TrickyDicky]

An interstellar medium enriched by stellar ejecta would be anything but static. A universe without entropy would be not even wrong, but, I naively suspect it would remain in its original state. A universe without BB nuclosynthesis still needs a source of hydrogen for primordial stars to form. Once the stellar formation process began, the ISM would be continuously enriched by stellar ejecta. I fail to see how that could ever achieve equilibrium. It would merely continue to be enriched until so heavily metallized it could no longer support stellar fusion. The answer seems to be stars would not form in a universe without entropy.

Exactly. This is merely a thought experiment. Objects like stars from our universe are not static- they age, things change, proportions of elements change with time. They couldn't exist in the form they have here. I find it hard to conceive of such a universe, the interrelationship of the laws that govern this universe are not workable without entropy. It is not that it is a physical law, merely that it is a consequence of the laws we operate under (in particular gravity).

You are thinking of primordial stars- our universe. The static universe TrickyDicky is presenting requires they just be here timelessly. An impossible requirement from my understanding of physics.
mathal

Of course it is just a thought experiment, like the many that are used in science to get a better understanding of things, when Einstein imagines he is riding a photon no-one thinks about the practical impossibility of doing that. Many things are explained in physics thru thought experiment or ideal models that are known not to exist in reality.
Anyway a static universe doesn't imply there is no "local" change, it is just the global cosmological parameters as a whole that don't change wrt time.
Otherwise it would have been plain silly from Einstein and the physicist of their time to even consider static spacetimes as models for our universe knowing as it is obvious that locally things change.

 

[Ken G]

Otherwise it would have been plain silly from Einstein and the physicist of their time to even consider static spacetimes as models for our universe knowing as it is obvious that locally things change.

Remember that when Einstein postulated the static universe, he did not even know there was such a thing as nuclear fusion. After the Hubble flow was found and fusion was discovered, both which happened a decade later, it was quickly realized that the Big Bang model explained the H/He ratio in a universe of finite age. It wasn't until later still that all the other nuclei got explained, when the role of stars was appreciated!

 

[TrickyDicky]

Remember that when Einstein postulated the static universe, he did not even know there was such a thing as nuclear fusion. After the Hubble flow was found and fusion was discovered, both which happened a decade later, it was quickly realized that the Big Bang model explained the H/He ratio in a universe of finite age. It wasn't until later still that all the other nuclei got explained, when the role of stars was appreciated!

We have total agreement on this.
Could you address the thought experiment?

 

[phyzguy]

We have total agreement on this.
Could you address the thought experiment?

Your thought experiment is basically, "In an imaginary universe where the known laws of physics do not apply, what would happen?"

The answer is, anything you want to happen. Since you have thrown the laws of physics out the window and made up your own laws of physics, we can't meaningfully speculate unless you tell us what laws of physics do apply.

 

[TrickyDicky]

Your thought experiment is basically, "In an imaginary universe where the known laws of physics do not apply, what would happen?"

The answer is, anything you want to happen. Since you have thrown the laws of physics out the window and made up your own laws of physics, we can't meaningfully speculate unless you tell us what laws of physics do apply.

Well, let's say that all the known physics would be the same except that the global entropy of such imaginary universe would be constant. Almost all physical laws are time reversible anyway.

 

[Chronos]

While physical laws are generally time reversible, but, we have no meaningful evidence of time reversed processes. This is why we observe and do experiments. Just because something is mathematically possible does not mean it is physically meaningful - e.g., quadratic equations have two solutions, but, both are not necessarily meaningful.

 

[Ken G]

We have total agreement on this.
Could you address the thought experiment?

I mean that in a static universe, with stellar nucleosynthesis, there is no steady-state H/He ratio-- there isn't any H or any He (it's all iron, the most stable nucleus). So if Einstein had known about stellar nucleosynthesis, he would have never suggested a static universe, and he would have been spared the embarrassment of missing the dynamical solution of his equations. Indeed this is my greatest puzzle about Einstein's model-- even what was known about stars at the time should have been enough to rule it out. It was already known that stars convert gravitational energy to light, and we've never seen anything that takes light and turns it back into gravitational energy in any significant way. But this is all related to the big mystery of what allowed stars to exist as we see them in the first place, which no one knew at the time.

 

[TrickyDicky]

I mean that in a static universe, with stellar nucleosynthesis, there is no steady-state H/He ratio-- there isn't any H or any He (it's all iron, the most stable nucleus).

I explained at least twice that my thought experiment was NOT an eternal steady-state universe.

 

[TrickyDicky]

I don't quite understand the resistance to even consider this thought experiment, when actually all the classical tests of relativity are computed using a similarly unrealistic model (even more because it is supposed to be an empty universe): the static exterior of a non-rotating star. I see no nitpicking in this case because everyone understands it is an exact solution of the EFE that allows a valid local approximation however unphysical the model looks.
Well, the OP imaginary universe is certainly no EFE solution, and of course I didn't expect anything valid for our universe to come out of it, but all thought experiments allow certain divergence from physical reality. That is why they are thought experiments.
Mathal was right that in such universe, without constraining it in any way, every distribution would be valid, that is why I asked if it was possible to apply the known stellar nuclear reactions and core conditions to single out some more probable equilibrium distribution. Maybe the problem is not well-posed to single out a certain distribution but so far nobody has said so.

 

[twofish-quant]

In static spacetimes nothing "ends up", there must be just an equilibrium distribution related to temperature, density and mass difference of protons and neutrons but independent of time.

In that case, everything turns into iron. The problem with that sort of universe is that it's pretty boring. Everything is iron. Stars don't exist.

Nuclear statistical equilibrium is pretty independent of temperature, density, and mass for normal temperatures. The equilibrium distributions will change only once you reach "nuclear" temperatures and densities. Anything less than that, it's 100% iron.

So now that question one has been set up, I'd like to make a universe that's slightly more interesting.

 

[twofish-quant]

I don't quite understand the resistance to even consider this thought experiment

There isn't. The problem is that I've mentioned the result of the thought experiment. Everything turns into iron. Once you have specified a density, temperature, and electron fraction, then there are lots of people that have calculated the "equilibrium state" of matter.

http://user.numazu-ct.ac.jp/~sumi/eos/

It will change for high temperatures and densities (i.e. T>1 million kelvin rho > 10^7 g/cm^2) but for anything under that, it's 100% iron nuclei. For high densities it start going to neutronium and maybe quark soup. For high temperatures, it starts turning into nucleon gas if it gets really hot.

I asked if it was possible to apply the known stellar nuclear reactions and core conditions to single out some more probable equilibrium distribution. Maybe the problem is not well-posed to single out a certain distribution but so far nobody has said so.

The answer is 100% iron.

Next question?

 

[TrickyDicky]

In that case, everything turns into iron. The problem with that sort of universe is that it's pretty boring. Everything is iron. Stars don't exist.

That answer is valid for a strictly time-dependent universe. You are not bothering to answer what I'm asking.

Nuclear statistical equilibrium is pretty independent of temperature, density, and mass for normal temperatures.

stellar core temperature is normal temperature to you?

 

[phyzguy]

Well, let's say that all the known physics would be the same except that the global entropy of such imaginary universe would be constant. Almost all physical laws are time reversible anyway.

As far as I can see, this is a meaningless statement. Perhaps you should study how the second law of thermodynamics comes about. It is a consequence of the large increase of the volume of phase space available near statistical equilibrium. I think any attempt to modify the laws of physics so that "the global entropy of such imaginary universe would be constant" would result in a universe that is unrecognizable. For example, I could say, "No interactions can occur". Well then, it's obvious the result of your static universe - whatever you start with stays in place forever.

 

[Ken G]

I explained at least twice that my thought experiment was NOT an eternal steady-state universe.

But that's the whole problem, that is precisely what your thought experiment is, since you acted as though the age of the universe is not a parameter in your question. There are only two possibilities-- either univeral age is not a relevant parameter, in which case you are talking about something "eternal", or else time is a dynamical parameter, in which case the answer will depend on the age. You did not say what that parameter was, thus you have to be talking about the former situation, there simply is no other possibility. Now, perhaps you mean that time is a parameter that has some understood value (like the usual 13.7 billion year age), but in addition to that parameter's value, there is also a kind of slowly varying quasi-steady solution that you are interested in. In that case, the problem is that the slowly varying quasi-steady value of H/He is pretty much just what we see, because in 13.7 billion years, stellar nucleosynthesis has not had time to have any real impact on the quasi-steady value of H/He (because not enough of the H is in massive enough stars to have an impact on H/He in that timescale). If you wait much longer, it will, but then H/He will be a function of age, and you have to say what age you have in mind. It will all be standard Big Bang, also.

 

[TrickyDicky]

But that's the whole problem, that is precisely what your thought experiment is, since you acted as though the age of the universe is not a parameter in your question. There are only two possibilities-- either univeral age is not a relevant parameter, in which case you are talking about something "eternal", or else time is a dynamical parameter, in which case the answer will depend on the age. You did not say what that parameter was, thus you have to be talking about the former situation, there simply is no other possibility. Now, perhaps you mean that time is a parameter that has some understood value (like the usual 13.7 billion year age), but in addition to that parameter's value, there is also a kind of slowly varying quasi-steady solution that you are interested in. In that case, the problem is that the slowly varying quasi-steady value of H/He is pretty much just what we see, because in 13.7 billion years, stellar nucleosynthesis has not had time to have any real impact on the quasi-steady value of H/He (because not enough of the H is in massive enough stars to have an impact on H/He in that timescale). If you wait much longer, it will, but then H/He will be a function of age, and you have to say what age you have in mind. It will all be standard Big Bang, also.

Anyone can look up easily in books or in wikipedia that a static spacetime is different than a steady-state universe. My thought experiment refers to a static one.

 

[Ken G]

Anyone can look up easily in books or in wikipedia that a static spacetime is different than a steady-state universe. My thought experiment refers to a static one.

Anyone, looking that up, would discover that all static universes are strict subsets of the class of all steady-state ones. That fact follows quite directly from the meanings of those words. As I said: you never gave an age. Now, is that because it doesn't matter? That is the definition of steady state.

 

[TrickyDicky]

Anyone, looking that up, would discover that all static universes are strict subsets of the class of all steady-state ones. That fact follows quite directly from the meanings of those words. As I said: you never gave an age. Now, is that because it doesn't matter? That is the definition of steady state.

The only explanation I can find to what you are saying is that you might be using the term "steady state" with a different meaning than I am. In fact in wikipedia at least two different meanings can be found: steady state as a kind of equilibrium of a system as used in many disciplines like thermodynamics and economics, and "steady state theory" or cosmology which is the specific model of universe that Hoyle et al. came up with in 1948 and that was seriously considered as alternative to BB universe until the 60's. This latter is the sense I have been giving to the term "steady state universe". It is well known that this model is that of an expanding universe. It is not possible therefore for static universes to be a subset of an expanding universe as I hope you will agree.
A a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field K which is irrotational, this is the standard definition and the one I'm following in my thought experiment as scenario for a putative plausible imaginary equilibrium distribution of chemical elements abundance.
Now, as was pointed out before, in abstract terms every distribution is compatible with such a universe. My question is, is there a way to constrain this with the known nuclear reactions (in reversible form) and the physical conditions of stellar's cores?

I thought this was an interesting exercise, I'm not so sure now.